Why is my half resolution render taking a quarter of the time of the full one?
That is simple math, area doesn't scale linearly, it has a quadratic relation to the dimensions of the shape, if you double the size you quadruple the area hence a four times the render time.

If the area of a $4 \times 4 = 16$ and you halve it then $2 \times 2 = 4$ which is one fourth

Hence you get a quarter of the area and consequently one fourth of pixels to calculate; that's about four times less computation power required leading to roughly one fourth of render time, give or take, for the same scene and settings.